When considering the variance of a portfolio with the returns of three assets $(R_A, R_B, R_C)$ and their respective weights $(\alpha, \beta, \gamma)$, the equation is given as:
$$ V(R) = \alpha^2 V(R_A) + \beta^2V(R_B) + \gamma^2V(R_C). $$
Could you please explain why the weights are squared in this calculation and explain the mathematical concepts behind this?
edit : (copy/paste from other thread that solved it for me)
$$ $$
$$\text{Var}(aX) = E[(aX)^2]-(E[aX])^2 = E[a^2 X^2]-(aE[X])^2 $$
$$=a^2 E[ X^2]-a^2(E[X])^2 $$ $$= a^2( E[X^2]-(E[X])^2 ) = a^2 \text{Var}(X)$$
edit : or this one may be more basic (depending on your definition of variance)
$$\text{Var}(aX) = E[(aX-E[aX])^2 ] = E[(aX-aE[X])^2 ] $$
$$=E[a^2(X-E[X])^2 ] $$ $$= a^2E[(X-E[X])^2 ] = a^2 \text{Var}(X)$$
The standard deviation $\sigma(\alpha R_A)$ is equal to $\alpha\,\sigma(R_A)$. And the variance is just the square of the standard deviation. So $$V(\alpha R_A)=\{\sigma(\alpha R_A)\}^2=\{\alpha\,\sigma(R_A)\}^2=\alpha^2\{\sigma(R_A)\}^2=\alpha^2 V(R_A)$$